Search Results (20)

We propose a universal framework for understanding system evolution based on structural complexity, offering a directional signature that applies across physical, chemical, and biological domains. Unlike entropy, which is constrained by its definition in closed, equilibrium systems, we introduce Kolmogorov Complexity (KC) and Fractal Dimension (FD) as quantifiable, scalable metrics that capture the emergence of organized complexity in open, non-equilibrium systems. We examine two major classes of systems: (1) living systems, revisiting Schrödinger’s insight that biological growth may locally reduce entropy while increasing structural order, and (2) irreversible natural processes such as oxidation, diffusion, and material aging. We formalize a Universal Law: expressed as a non-decreasing function Ω(t) = α·KC(t) + β·FD(t), which parallels the Second Law of Thermodynamics but tracks the rise in algorithmic and geometric complexity. This framework integrates principles from complexity science, providing a robust, mathematically grounded lens for describing the directional evolution of systems across scales-from crystals to cognition. Full article

The ATLAZUL project is an Interreg effort among 18 partners from Spain and Portugal along the Atlantic Iberian coasts. One of its objectives is the development of new methods and data processing for oceanic information to produce useful products for private and public stakeholders. This study proposes a new insight on the sea surface dynamic of the ATLAZUL area based on almost two years of multiscale high resolution sea surface temperature imagery. The use of techniques such as the Karhunen–Loève transform (Empirical Orthogonal Function) and the Maximum Entropy Spectral Analysis were applied to study long- and short-term features in the sea surface temperature imagery. Mathematical Morphology and the Geometrical Theory of Measure are utilized to compute the Medial Axis Transform and the Hausdorff dimension. The results can be summarized as follows: (i) the tow upwelling areas are identified along the Galician–Portugal coast as indicated in the second and third modes of KLT/EOF analysis, and they are partially affected by wind; (ii) the tow warm water outflows from the Bay of Cádiz to the Gulf of Cádiz are identified as the second and third modes of KLT/EOF analysis, which are also influenced by wind; (iii) the skeletons of the surface signature of the upwelling and of the warmer water outflow, along with their fractal dimensions, indicate a chaotic pattern of spatial distribution and (iv) the harmonic prediction model should be combined with the wind prediction. Full article

In this paper, a novel class of rational cubic fractal interpolation function (RCFIF) has been proposed, which is characterized by one shape parameter and a linear denominator. In interpolation for shape preservation, the proposed rational cubic fractal interpolation function provides a simple but effective approach. The nature of shape preservation of the proposed rational cubic fractal interpolation function makes them valuable in the field of data visualization, as it is crucial to maintain the original data shape in data visualization. Furthermore, we discussed the upper bound of error and explored the mathematical framework to ensure the convergence of RCFIF. Shape parameters and scaling factors are constraints to obtain the desired shape-preserving properties. We further generalized the proposed RCFIF by introducing the concept of signature, giving its construction in the form of a zipper-rational cubic fractal interpolation function (ZRCFIF). The positivity conditions for the rational cubic fractal interpolation function and zipper-rational cubic fractal interpolation function are found, which required a detailed analysis of the conditions where constraints on shape parameters and scaling factor lead to the desired shape-preserving properties. In the field of shape preservation, the proposed rational cubic fractal interpolation function and zipper fractal interpolation function both represent significant advancement by offering a strong tool for data visualization. Full article

A new instrument using reflectance transformation imaging (RTI), named MorphoLight, has been developed for surface characterization. This instrument is designed to be adjustable to surfaces, ergonomic, and uses a combination of high-resolution imaging functions, i.e., focus stacking (FS) and high dynamic range (HDR), to improve the image quality. A topographical analysis method is proposed with the instrument. This method is an improvement of the surface gradient characterization by light reflectance (SGCLR) method. This aims to analyze slope/curvature maps, traditionally studied in RTI, but also to find the most relevant lighting position and 3D surface parameter which highlight morphological signatures on surfaces and/or discriminate surfaces. RTI measurements and analyses are performed on two zones, sky and sea, of a naval painting which have the same color palette but different painting strokes. From the statistical analysis using bootstrapping and analysis of variance (ANOVA), it is highlighted that the high-resolution images (stacked and tonemapped from HDR images) improve the image quality and make it possible to better see a difference between both painting zones. This difference is highlighted by the fractal dimension for a lighting position (θ, φ) = (30°, 225°); the fractal dimension of the sea part is higher because of the presence of larger brushstrokes and painting heaps. Full article

Entropy-based and fractal-based metrics derived from heart rate variability (HRV) have enriched the way cardiovascular dynamics can be described in terms of complexity. The most commonly used multifractal testing, a method using q moments to explore a range of fractal scaling in small-sized and large-sized fluctuations, is based on detrended fluctuation analysis, which examines the power–law relationship of standard deviation with the timescale in the measured signal. A more direct testing of a multifractal structure exists based on the Shannon entropy of bin (signal subparts) proportion. This work aims to reanalyze HRV during cognitive tasks to obtain new markers of HRV complexity provided by entropy-based multifractal spectra using the method proposed by Chhabra and Jensen in 1989. Inter-beat interval durations (RR) time series were obtained in 28 students comparatively in baseline (viewing a video) and during three cognitive tasks: Stroop color and word task, stop-signal, and go/no-go. The new HRV estimators were extracted from the f/α singularity spectrum of the RR magnitude increment series, established from q-weighted stable (log–log linear) power laws, namely: (i) the whole spectrum width (MF) calculated as α_max − α_min; the specific width representing large-sized fluctuations (MF_large) calculated as α₀ − α_q+; and small-sized fluctuations (MF_small) calculated as α_q− − α₀. As the main results, cardiovascular dynamics during Stroop had a specific MF signature while MF_large was rather specific to go/no-go. The way these new HRV markers could represent different aspects of a complete picture of the cognitive–autonomic interplay is discussed, based on previously used entropy- and fractal-based markers, and the introduction of distribution entropy (DistEn), as a marker recently associated specifically with complexity in the cardiovascular control. Full article

With the increasing exploitation of mineral resources by humans, exploring non-traditional areas for hidden resources such as deep earth and sediment-covered regions has become a significant challenge in the field of mineral exploration. Geochemical data, as a crucial information carrier of geological bodies, serves as one of the direct and effective sources for quantitative analysis of regional geological evolution and mineralization prediction studies. It plays an indispensable role in geographic information system (GIS)-based mineral exploration. Due to the neglect of spatial distribution characteristics and the variability of statistical features with spatial metrics in traditional statistical methods, this paper employs fractal/multifractal and the local singularity analysis to identify geochemical anomalies from background and characterize geochemical distributions associated with porphyry Cu-Au mineralization in the Duolong mineral district, Tibet, China. A novel algorithm for estimating the singularity index, which takes anisotropy into consideration, is proposed and practically applied to the Duolong district. By comparing with the isotropic singularity index, this new method objectively identifies anisotropic geochemical signatures and investigates non-linear behaviors of ore-forming elements, making it more practical and effective in geo-anomaly extraction. Furthermore, the current method is capable of indicating variations in geochemical distributions at different scales through directional arrows marking analytical windows. The summed-up direction of these multi-scale vectors effectively demonstrates migration trends of ore materials at each location within the study area. The new method can pinpoint the location of ore-forming element accumulation and migration directions, unlocking valuable insights from complex datasets. This promises to revolutionize our understanding of how minerals are formed and distributed within the Earth’s crust. Full article

A class of zipper fractal functions is more versatile than corresponding classes of traditional and fractal interpolants due to a binary vector called a signature. A zipper fractal function constructed through a zipper iterated function system (IFS) allows one to use negative and positive horizontal scalings. In contrast, a fractal function constructed with an IFS uses positive horizontal scalings only. This article introduces some novel classes of continuously differentiable convexity-preserving zipper fractal interpolation curves and surfaces. First, we construct zipper fractal interpolation curves for the given univariate Hermite interpolation data. Then, we generate zipper fractal interpolation surfaces over a rectangular grid without using any additional knots. These surface interpolants converge uniformly to a continuously differentiable bivariate data-generating function. For a given Hermite bivariate dataset and a fixed choice of scaling and shape parameters, one can obtain a wide variety of zipper fractal surfaces by varying signature vectors in both the x direction and y direction. Some numerical illustrations are given to verify the theoretical convexity results. Full article

Amyotrophic lateral sclerosis (ALS) is a fatal neurodegenerative disease whose diagnosis depends on the presence of combined lower motor neuron (LMN) and upper motor neuron (UMN) degeneration. LMN degeneration assessment is aided by electromyography, whereas no equivalent exists to assess UMN dysfunction. Magnetic resonance imaging (MRI) is primarily used to exclude conditions that mimic ALS. We have identified four different clinical/radiological phenotypes of ALS patients. We hypothesize that these ALS phenotypes arise from distinct pathologic processes that result in unique MRI signatures. To our knowledge, no machine learning (ML)-based data analyses have been performed to stratify different ALS phenotypes using MRI measures. During routine clinical evaluation, we obtained T1-, T2-, PD-weighted, diffusion tensor (DT) brain MRI of 15 neurological controls and 91 ALS patients (UMN-predominant ALS with corticospinal tract CST) hyperintensity, n = 21; UMN-predominant ALS without CST hyperintensity, n = 26; classic ALS, n = 23; and ALS patients with frontotemporal dementia, n = 21). From these images, we obtained 101 white matter (WM) attributes (including DT measures, graph theory measures from DT and fractal dimension (FD) measures using T1-weighted), 10 grey matter (GM) attributes (including FD based measures from T1-weighted), and 10 non-imaging attributes (2 demographic and 8 clinical measures of ALS). We employed classification and regression tree, Random Forest (RF) and also artificial neural network for the classifications. RF algorithm provided the best accuracy (70–94%) in classifying four different phenotypes of ALS patients. WM metrics played a dominant role in classifying different phenotypes when compared to GM or clinical measures. Although WM measures from both right and left hemispheres need to be considered to identify ALS phenotypes, they appear to be differentially affected by the degenerative process. Longitudinal studies can confirm and extend our findings. Full article

A growing fatigue crack in metallic materials and structures exhibits multifractal features that inherit signatures of the crack growth rate behavior of the material. This study exploits the recently established multifractal fatigue crack growth model to quantify the characteristic fatigue crack growth rate response of the AISI 410 martensitic stainless steel using an L-shaped bell crank structure. The objective is to demonstrate that the fatigue crack growth rate response of the material could be established by quantifying the fractality of the growing crack. The fractal approach avoids the need of the crack geometry factor when calculating the crack tip driving force. The fractal analysis of the crack image employs the box-counting algorithm to determine the fractal dimension along the edge of the crack length. The analysis is confined to the power law crack growth rate stage (Paris crack growth regime). Results show that the fatigue crack growth path in the bell crank structure is dictated by the Mode I (opening) component of the crack loading. The distribution of fractal-based fatigue crack growth rate data is within the 99% confidence limit of the median crack growth response by the Paris equation. Thus, the model could be employed for prediction of the fatigue crack growth response of engineering structures where the crack geometry factor is not readily available. Full article

Since computing advances in the last 30 years have allowed automated calculation of fractal dimensions, fractals have been established as ubiquitous signatures of urban form and socioeconomic function. Yet, applications of fractal concepts in urban planning have lagged the evolution of technical analysis methods. Through a narrative literature review around a series of “big questions” and automated bibliometric analysis, we offer a primer on fractal applications in urban planning, targeted to urban scholars and participatory planners. We find that developing evidence demonstrates linkages between urban history, planning context, and urban form and between “ideal” fractal dimension values and urban aesthetics. However, we identify gaps in the literature around findings that directly link planning regulations to fractal patterns, from both positive and normative lenses. We also find an increasing trend of most literature on fractals in planning being published outside of planning. We hypothesize that this trend results from communication gaps between technical analysts and applied planners, and hope that our overview will help to bridge that gap. Full article

The two most significant signatures of the Anthropocene—agriculture and urbanisation—have yet to be studied synoptically. The term periurban is used to describe territory where the urbanising trend of the planet extends into multiscapes. A periurban praxis is required that spatially reconciles urbanisation and agriculture, simultaneously permitting urban growth and the enhancement of critical ecosystem services provided by agricultural hinterlands. This paper presents a synthesis of four fields of ecological research that converge on periurban multiscapes—ecological urbanism, landscape ecology, ecosystem services science and agroecology. By applying an ecosystem services approach, a diagram is developed that connects these fields as a holistic praxis for spatially optimising periurban multiscapes for ecosystem services performance. Two spatial qualities of agroecology—‘ES Density’ and ‘ES Plasticity’—potentiate recent areas of research in each of the other three fields—ecology for the city from ecological urbanism, landscape metrics from landscape ecology (particularly the potential application of fractals and surface metrics) and ecosystem services supply and demand mapping and ‘ES Space’ theory from ecosystems services science. While the multifunctional value of agroecological systems is becoming widely accepted, this paper focuses on agroecology’s specific spatial value and its unique capacity to supply ecosystem services specifically tailored to the critical ecosystemic demands of periurban multiscapes. Full article

Recently, a variety of intelligent structural damage identification algorithms have been developed and have obtained considerable attention worldwide due to the advantages of reliable analysis and high efficiency. However, the performances of existing intelligent damage identification methods are heavily dependent on the extracted signatures from raw signals. This will lead to the intelligent damage identification method becoming the optimal solution for actual application. Furthermore, the feature extraction and neural network training are time-consuming tasks, which affect the real-time performance in identification results directly. To address these problems, this paper proposes a new intelligent data fusion system for damage detection, combining the probabilistic neural network (PNN), data fusion technology with correlation fractal dimension (CFD). The intelligent system consists of three modules (models): the eigen-level fusion model, the decision-level fusion model and a PNN classifier model. The highlight points of this system are these three intelligent models specialized in certain situations. The eigen-level model is specialized in the case of measured data with enormous samples and uncertainties, and for the case of confidence level of each sensor is determined ahead, the decision-level model is the best choice. The single PNN model is considered only when the data collected is somehow limited, or few sensors have been installed. Numerical simulations of a two-span concrete-filled steel tubular arch bridge in service and a seven-storey steel frame in laboratory were used to validate the hybrid system by identifying both single- and multi-damage patterns. The results show that the hybrid data-fusion system has excellent performance of damage identification, and also has superior capability of anti-noise and robustness. Full article

We propose a carbon-nanotube-based neural sensor designed to exploit the electrical sensitivity of an inhomogeneous fractal network of conducting channels. This network forms the active layer of a multi-electrode field effect transistor that in future applications will be gated by the electrical potential associated with neuronal signals. Using a combination of simulated and fabricated networks, we show that thin films of randomly-arranged carbon nanotubes (CNTs) self-assemble into a network featuring statistical fractal characteristics. The extent to which the network’s non-linear responses will generate a superior detection of the neuron’s signal is expected to depend on both the CNT electrical properties and the geometric properties of the assembled network. We therefore perform exploratory experiments that use metallic gates to mimic the potentials generated by neurons. We demonstrate that the fractal scaling properties of the network, along with their intrinsic asymmetry, generate electrical signatures that depend on the potential’s location. We discuss how these properties can be exploited for future neural sensors. Full article

Bone fractal signature analysis (FSA—also termed bone texture analysis) is a tool that assesses structural changes that may relate to clinical outcomes and functions. Our aim was to compare bone texture analysis of the distal radius in patients and volunteers using radiography and 3T and 7T magnetic resonance imaging (MRI)—a patient group (n = 25) and a volunteer group (n = 25) were included. Participants in the patient group had a history of chronic wrist pain with suspected or confirmed osteoarthritis and/or ligament instability. All participants had 3T and 7T MRI including T1-weighted turbo spin echo (TSE) sequences. The 7T MRI examination included an additional high-resolution (HR) T1 TSE sequence. Radiographs of the wrist were acquired for the patient group. When comparing patients and volunteers (unadjusted for gender and age), we found a statistically significant difference of horizontal and vertical fractal dimensions (FDs) using 7T T1 TSE-HR images in low-resolution mode (horizontal: p = 0.04, vertical: p = 0.01). When comparing radiography to the different MRI sequences, we found a statistically significant difference for low- and high-resolution horizontal FDs between radiography and 3T T1 TSE and 7T T1 TSE-HR. Vertical FDs were significantly different only between radiographs and 3T T1 TSE in the high-resolution mode; FSA measures obtained from 3T and 7T MRI are highly dependent on the sequence and reconstruction resolution used, and thus are not easily comparable between MRI systems and applied sequences. Full article

Social systems are characterized by an enormous network of connections and factors that can influence the structure and dynamics of these systems. Among them the whole economical sphere of human activity seems to be the most interrelated and complex. All financial markets, including the youngest one, the cryptocurrency market, belong to this sphere. The complexity of the cryptocurrency market can be studied from different perspectives. First, the dynamics of the cryptocurrency exchange rates to other cryptocurrencies and fiat currencies can be studied and quantified by means of multifractal formalism. Second, coupling and decoupling of the cryptocurrencies and the conventional assets can be investigated with the advanced cross-correlation analyses based on fractal analysis. Third, an internal structure of the cryptocurrency market can also be a subject of analysis that exploits, for example, a network representation of the market. In this work, we approach the subject from all three perspectives based on data from a recent time interval between January 2019 and June 2020. This period includes the peculiar time of the Covid-19 pandemic; therefore, we pay particular attention to this event and investigate how strong its impact on the structure and dynamics of the market was. Besides, the studied data covers a few other significant events like double bull and bear phases in 2019. We show that, throughout the considered interval, the exchange rate returns were multifractal with intermittent signatures of bifractality that can be associated with the most volatile periods of the market dynamics like a bull market onset in April 2019 and the Covid-19 outburst in March 2020. The topology of a minimal spanning tree representation of the market also used to alter during these events from a distributed type without any dominant node to a highly centralized type with a dominating hub of USDT. However, the MST topology during the pandemic differs in some details from other volatile periods. Full article